Everton, their players re-jigged to fit a 4-2-3-1 formation, understandably took time to build momentum during a low-key opening quarter.
The match awoke from its slumber in the 17th minute when a challenge by Billy Jones on Anichebe saw the ball ricochet off the Everton man and drop just wide of former Wrexham goalkeeper Ben Foster’s goal.
Two minutes later, a long diagonal pass from Baines was headed down by Sylvain Distin into the path of Marouane Fellaini, who took the ball on his chest and then swivelled a shot against the outside of the post.
Baines then unlocked the Baggies defence in emphatic fashion on 29 minutes. Continuing his run infield from the left flank, the defender dashed beyond Jones and Jonas Olsson on to Leon Osman’s pass before smashing a shot from 20 yards past West Brom goalkeeper Foster.
The visitors were disrupted further two minutes before the interval when George Thorne landed awkwardly after challenging with Anichebe and, after lengthy treatment, left the field on a stretcher.
His replacement, Claudio Yacob, then suffered a disastrous introduction by bowling over Anichebe inside the area in the first minute of stoppage time. Baines sent Foster the wrong way from the spot; he has yet to miss a penalty in normal playing time for Everton.
With Fellaini and Phil Neville locking down central midfield, West Brom’s dangerous, powerful forward, Romelu Lukaku, was starved of decent service, the visitors unable to utilise his pace on the counter-attack.
Everton remained in control after the interval. Mirallas cut inside off the left flank and fired a right-footed shot that required a parry from Foster, while the Baggies goalkeeper then clutched a trademark curling Pienaar effort.
So it was out of nothing when West Brom were gifted a way back into the match on 65 minutes.
Lukaku was played into space by Chris Brunt and went around Tim Howard, but his shot from the angle lacked pace. However, Heitinga attempted to control the ball rather than boot it clear, overcooked his touch and allowed Long, only on the pitch three minutes, the chance to head in.
The Baggies were invigorated and, moments later, Graham Dorrans smashed the crossbar with a scintillating free-kick from 25 yards.
Everton responded. Foster saved bravely at the feet of Osman while Pienaar later forced the goalkeeper to turn the ball over with a curling free-kick after the omnipresent Baines had been fouled.
Osman then missed a gilt-edged chance to ease Everton nerves when he guided a shot over with the goal gaping after Pienaar had struck the post.
Moyes’ team, though, held on. And with Baines leading the way, their determined celebrations at the final whistle indicate they believe they can go fourth and conquer.
MAN OF THE MATCH. Leighton Baines. Two goals and yet another excellent display from the in-form defender.